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Noise sensitivity in continuum percolation. (English) Zbl 1305.60100
Authors’ abstract: We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first which involves a percolation model with critical probability \(p_c = 1/2\). Our proof uses a version of the Benjamini-Kalai-Schramm theorem for biased product measures. A quantitative version of this result was recently proved by N. Keller and G. Kindler [Combinatorica 33, No. 1, 45–71 (2013; Zbl 1299.05308)]. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete models with pc bounded away from zero; this method is based on an extremal result for non-uniform hypergraphs.
Reviewer’s remark: The reading of this advanced contribution presupposes a profound knowledge of the contents of the monographs [N. Alon and J. H. Spencer, The probabilistic method. With an appendix on the life and work of Paul Erdős. 3rd ed. Hoboken, NJ: John Wiley & Sons (2008; Zbl 1148.05001); B. Bollobas and O. Riordan, Percolation. Cambridge: Cambridge University Press (2006; Zbl 1118.60001); G. Grimmett, Percolation. 2nd ed. Berlin: Springer (1999; Zbl 0926.60004); R. Meester and R. Roy, Continuum percolation. Cambridge: Cambridge Univ. Press (1996; Zbl 0858.60092)].

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
94C10 Switching theory, application of Boolean algebra; Boolean functions (MSC2010)
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